Consider a down-and-in American call C di (S, ; X,B), where the down-and-in trigger clause entitles

Consider a down-and-in American call C_di (S, τ ; X,B), where the down-and-in trigger clause entitles the holder to receive an American call option with strike price X when the asset price S falls below the threshold level B. The underlying asset pays dividend yield q and let r denote the riskless interest rate. Let C(S,τ ; X) and c(S,τ ; X) denote the price function of the American call and European call with strike price X, respectively. Show that when B ≤ max (X, r/q X), we have

where c_di (S, τ ; X,B) is the price function of the European down-and-in call counterpart. Find the corresponding form of the price function C_di (S, τ ; X,B) when (i) B ≥ S^∗(∞) and (ii) S^∗(0+) ∗(∞), where S^∗(τ) is the optimal exercise boundary of the American non-barrier call C(S,τ ; X) (Dai and Kwok, 2004). 

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Cdi (S, T, X, B) 2(r-q) 1- B2 " [C(B ²7. T;X) - C( B²₁, T; X )] S S S = (-²/ B +Cdi (S, T; X, B),